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Distributed Optimization in Large Interconnected Systems using ADMM

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This thesis focuses on the construction of distributed algorithms for optimizing resource production in a large interconnected system. In particular, it focuses on power grid and 5G cellular networks.

In the case of power grid networks, we consider the OPF (Optimal Power Flow) problem in which one seeks to manage and optimize the production of electrical energy in a distributed manner. We focus on a linearized version of the problem, the DC-OPF (Direct-Current Optimal Power Flow) problem. This optimization problem is convex; the aim is to minimize the cost of energy generation while respecting the limits of the transmission line and the power constraints.

In the case of 5G cellular networks, we formulate a caching problem. We aim to offload the backhaul link usage connecting the small bases stations (SBSs) to the central scheduler (CS). The SBSs are equipped with a limited storage capacity. We seek to find the optimal way to store files so as to reduce the cost on the use of backhaul and sharing files with other SBSs.

The approach adopted in this thesis is to apply the ADMM (Alternating Direction Method of Multipliers), an optimization method that is applied iteratively, to an optimization problem that we adequately formulated previously. This problem can both describe the DC-OPF problem and the Caching problem. We prove the convergence of the method when applied node by node in a fully distributed manner. Additionally, we prove its convergence in the case where the network is divided into multiple areas or nations that may or may not overlap. Furthermore, in the context of a network with multiple areas, we show that the application of ADMM in a random manner by a single randomly chosen area also converges to the optimal solution of the problem.